Commun. Theor. Phys. (Beijing, China) 43 (2005) pp. 159–164
c International Academic Publishers
Vol. 43, No. 1, January 15, 2005
Theoretical Studies on Photoionization Cross Sections of Solid Gold∗
MA Xiao-Guang, SUN Wei-Guo,† and CHENG Yan-Song
Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, China
(Received May 9, 2004)
Abstract Accurate expression for photoabsorption (photoionization) cross sections of high density system proposed
recently is used to study the photoionization of solid gold. The results show that the present theoretical photoionization
cross sections have good agreement both in structure and in magnitude with the experimental results of gold crystal.
The studies also indicate that both the real part ε0 and the imaginary part ε00 of the complex dielectric constant ε,
and the dielectric influence function of a nonideal system have rich structures in low energy side with a range about
50 eV, and suggest that the influence of particle interactions of surrounding particles with the photoionized particle on
the photoionization cross sections can be easily investigated using the dielectric influence function. The electron overlap
effects are suggested to be implemented in the future studies to improve the accuracy of theoretical photoionization cross
sections of a solid system.
PACS numbers: 32.80.Fb, 77.22.-d, 78.20.Ci
Key words: photoionization cross section, solid gold, dielectric influence function
1 Introduction
Photoionization and photodetachment processes are
the simplest electron transfer reactions induced by photoabsorption. Understanding of these processes and the
related properties is fundamental and important to many
physical and chemical studies. For example, photoionization data are necessary inputs for modelling phenomena
in plasmas and neutral fluids, etc.[1] Most of these studies
address only on atoms/molecules in ideal gas, or on single chain oligomers or small clusters of molecules without
relating to macroscopic systems.[2] Recently, some photoionization experiments have aimed at studying the characteristics and properties of solid-state, oligomers, and
clusters,[3] the temperature dependence of photoionization
cross sections,[4] the dynamic processes of photoionization
of liquid,[5] and the atomic/molecular interactions of the
photoionization processes,[6] rather than studying single
atom. Phenomena such as line broadening and spectral
shifts of the optical spectra of atoms in condensed states
contain information on both the dynamics of liquid and
the interatomic interactions. Many measured properties
of atomic and molecular photoionization stem from experiments in some condensed phases. Photoionization cross
sections of atoms/molecules in condensed phase may be
sensitive to the “environmental” effects — the electromagnetic effects of surrounding particles and the interactions among particles. Understanding the relation between macroscopic electromagnetic responses observed in
experiments and microscopic properties such as atomic
and molecular polarizabilities still presents a challenge.[7]
Recent theoretical study provides a basis for identifying
the influence of the state of aggregation (the number density effects, etc.) on the observable properties of matter.[8]
The present work investigates the number density effect on photoionization cross sections of atoms in real
system using our alternative expression for photoionization cross section and dielectric influence functions. The
interatomic interactions are considered using the wellknown Clausius–Mossotti model.[9] The theoretical expression for photoionization cross sections of condensed
matter is based on experimental Beer–Lambert’s law,[10]
which is derived directly from the classical response of
the condensed system and is independent of any quantum
mechanical method. The theoretical photoionization cross
sections of solid gold are compared with the experimental
values of both ideal gas and solid phase. The results show
that the suggested method is reliable and would open a
new way to calculate the photoionization cross sections of
complex molecules, clusters, and solid system.
2 Photoionization Cross Sections of
Condensed Phase
In a photoabsorption experiment, the photoabsorption
cross sections are determined by detecting the light attenuation through a medium. It is known that the number
density of the medium significantly affects the measured
cross section. The photoabsorption (when quantum efficiency δ = 1, photoionization) cross section σ at photon
energy ω is determined by the Beer–Lambert’s law,[10]
1 I0 σ=
ln
,
(1)
Nl
I
∗ The project supported by National Natural Science Foundation of China under Grant No. 10474068 and the Science Foundation of the
Ministry of Education of China
† Correspondence author, E-mail: [email protected]; [email protected]
160
MA Xiao-Guang, SUN Wei-Guo, and CHENG Yan-Song
where I0 is the intensity of the incident light, I is the attenuated intensity of the transmitted light, l is the transmitted length of light, and N is the number density of the
system. Unless indicated atomic units are used throughout. Based on the above Beer–Lambert’s law of Eq. (1),
and using the first order approximation in a series expansion on the dielectric constant in an ideal gas model,
Fano and Cooper[11] gave the expression for photoionization cross sections σi of an isolated atom or molecule used
in most theoretical studies.[12−15]
4πω
ζ,
(2)
σi (ω) =
c
where c is the speed of light, and ζ is the imaginary
part of dynamic dipole polarizability α(η + iζ) of a free
atom/molecule. σi in Eq. (2) is independent of any macroscopic quantity of a system, and represents the property
of an isolated atom. Equation (2) is valid usually at the
range of room temperature (273 K) and pressure (1 atm),
σ(ω) =
where
Vol. 43
under which the number density of particles
7 −3
N = 3.99 × 10−6 a−3
0 = 2.70 × 10 µ
(a0 is the Bohr radius and µ, micron) for ideal gas and
the interatomic interactions can be completely neglected.
However, photoionization cross section of atoms in a
real system may be sensitive to “environmental” effect
which might be represented by some macroscopic properties such as the complex dielectric constant ε of the system. In order to relate the microscopic quantity (cross
section) of Eq. (2) to some macroscopic quantities of a
photoionization system, we have proposed an alternative
expression[16] for photoionization cross section of atoms in
real system based on Beer–Lambert’s law without using
any mathematical approximation and physical model,
√ q
2ω p 02
ε + ε002 − ε0 ,
(3)
σ(ω) =
Nc
which can be rewritten in a series expansion for the case
of ε00 < ε0 as
∞
002 m−1 io1/2
X
ω 00 n 1 h
ωε00
m+1 2(2m − 3)!! ε
ε
1
+
(−1)
f (ε0 , ε00 ) ,
=
02
Nc
ε0
(2m)!!
ε
N
c
m=2
2
∞
X
1 h
2(2m − 3)!! ε00 m−1 i1/2
f (ε0 , ε00 ) = √ 1 +
(−1)m+1
,
(2m)!!
ε02
ε0
m=2
(4)
(5)
while ε0 and ε00 are the real part and the imaginary part of the complex dielectric constant ε(= ε0 + iε00 ) of a macrosystem
respectively. Comparing Eqs. (2) and (3), one can see that equation (3) is a function of macroscopic properties (ε, N )
and can be used to evaluate the cross sections of both condensed phase and ideal gas systems, while equation (2) is
only the function of microscopic property ζ of a single particle and can only describe the cross sections of an isolated
particle. Equation (3) is a strict expression and gives a convenient way to understand the relation between macroscopic
electromagnetic responses such as the dielectric constant (ε0 , ε00 ) observed in experiments and microscopic properties
such as atomic, and molecular polarizabilities α(= η + iζ). On the right-hand side of Eq. (3), the complex dielectric
constant represents the statistic average of the macroscopic dielectric property which includes the influence of other
particles in the environment on the ionized atom, while on the left-hand side is the microscopic quantum quantity.
Equation (3) can also be rewritten as[16]
qp
1
4πω
ζ· √
ε02 + ε002 − ε0 = σi (ω) · f (N, ε0 , ε00 ; ζ)
(6)
σ(ω) =
c
2 2 πN ζ
with
1
f (N, ε , ε ; ζ) = √
2 2 πN ζ
0
00
qp
ε02 + ε002 − ε0 .
(7)
Equation (6) indicates that the photoionization cross sections for the atom or molecule in the real system can be
obtained by transforming the photoionization cross section σi of the same particle in ideal gas using a transforming
function f (N, ε0 , ε00 ; ζ). Since f (N, ε0 , ε00 ; ζ) is the function of the number density N and the dielectric constant (ε0 , ε00 )
of the system, it may be called as Dielectric Influence Function (DIF). For dilute and ideal gas, the number density N
is so small that there are almost no interactions among particles, and DIF → 1 and equation (2) becomes the accurate
expression for photoionization cross sections. However, as the number density of the system increases, the influence of
the dielectric property on cross section becomes more and more important, and the interactions among atoms are not
negligible for high density gas, liquid or solid. In this situation the DIF is not near unity in some energy range, and it
includes not only higher power dielectric terms but also the interactions among atoms.
No. 1
Theoretical Studies on Photoionization Cross Sections of Solid Gold
161
For the case of polarized atoms/molecules arranged in cubic crystals, the relation between the microscopic polarizability α(ω) [= η(ω) + iζ(ω)] and the macroscopic dielectric function ε(ω) [= ε0 (ω) + iε00 (ω)] can be obtained using
the well-known Clausius–Mossotti (CM) model as[16]
ε0 =
(3 + 8πN η)(3 − 4πN η) − 32(πN ζ)2
,
(3 − 4πN η)2 + (4πN ζ 2 )
(8)
ε00 =
36πN ζ
.
(3 − 4πN η)2 + (4πN ζ)2
(9)
Equations (8) and (9) describe the collective influence of N particles having the same microscopic polarizability α(ω) =
η(ω) + iζ(ω) on the dielectric constant ε(ω) = ε0 (ω) + iε00 (ω) of a macrosystem with cubic crystal structure. For most
ideal gas, N is in the order of 10−6 , 4πN is in 10−5 ; while for most real gas, N and 4πN are in 10−4 and 10−3
respectively; and for most cubic crystals, N and 4πN are in 10−3 and 10−2 respectively. The DIF f (N, ε0 , ε00 ; ζ) in
Eq. (6) may have different forms for photoionization systems in different physical phases or models. When CM model
is used for cubic crystals, or liquid system, or real gas, f (N, ε0 , ε00 ; ζ) would have the following form[16]
∞
X
1 h
9
2(2m − 3)!! ε002 m−1 i1/2
√
1
+
(−1)m+1
.
(10)
fCM (N, ε0 , ε00 ; ζ) =
2
2
(3 − 4πN η) + (4πN ζ)
(2m)!!
ε02
ε0
m=2
This fCM (N, ε0 , ε00 ; ζ) contains both particle interactions and all higher power terms in the expansion of ε1/2 . Therefore,
the photoionization cross sections of the cubic solid or real gas system can be obtained through Eq. (6) by transforming
the ideal gas cross sections σi in Eq. (2) using the DIF in Eq. (10).
3 Applications
Using the accurate experimental[17] photoionization cross sections of isolated atomic gold, one can obtain the
imaginary part ζ of the polarizability of gold atom using Eq. (2), and performs a Kramers–Kronig transformation[18]
to obtain the real part η of the frequency-dependent polarizability from the known ζ. For real system, the real part
ε0 and the imaginary part ε00 of the dielectric constant can be generated using Eqs. (8) and (9) from (η, ζ). Figure 1
shows the relative magnitudes of the real and imaginary parts of dielectric functions of gold atom and their variations
with both the photon frequency and the number density N . It can be seen that both ε0 and ε00 increase as the number
density N increases, and that the real part ε0 has a broad peak with structures at lower photon energies for each of
the higher particle densities, and starts to decrease monotonically at 90 eV. The imaginary part ε00 has a narrower but
stronger peak than that of ε0 at lower energies for each density, and starts to decrease at 40 eV and finally go to a
small value near zero.
Fig. 1 The real part ε0 and the imaginary part ε00 of dielectric constant of atomic gold vary with photon energies at
different number densities.
162
MA Xiao-Guang, SUN Wei-Guo, and CHENG Yan-Song
Vol. 43
Figure 2 shows the CM dielectric influence functions (DIF) of gold atom at some number densities with the lowest
−6 −3
Nl = 3.99 × 10−6 a0−3 and the highest Nh = 7.72 × 10−3 a−3
a0 of the
0 . It is shown that at the density N = 3.99 × 10
ideal gas, the DIF fCM is close to unity at energies below 80 eV, and becomes one for ω > 80 eV. This fact indicates that
the DIF has no effect on photoionization cross sections of ideal gas. As the density increases, the higher the density, the
stronger the structure of the DIF fCM is for photon energies below 110 eV. The DIF structure monotonically decreases
from 90 eV for all high particle densities, and becomes to a constant near unity at 250 eV. Therefore, the DIF has
greater affections as shown in Eq. (6) on the photoionization cross sections of high density systems such as cubic
crystals at energies below 110 eV, and the affection is gradually diminishing from 90 eV and finally vanishes in high
energy regions. As a result, the photoionization cross sections of cubic crystals will almost be the same as those of ideal
gas in high energies. This example indicates that the influence of both the particle interactions and the macroscopic
dielectric properties on the photoionization cross sections of a high density system can be measured by the DIF.
Fig. 2
The dielectric influence function of atomic gold varies with photon energies at different number densities.
Figure 3 shows the comparison of the gold photoionization cross sections among the solid phase experiment (“ooo”)[19] determined by use of synchrotron radiation on thin films, the ideal gas experimental results
σi (“ · · · · · ·”)[17] obtained from the photoabsorption experiments, and the present theoretical results (“—”) calculated with the solid phase number density N = 8.72 ×
10−3 a−3
at room temperature. The present theoretical
0
photoionization cross sections for solid gold are generated
by transforming the ideal gas photoionization cross sections σi using Eq. (6) and the DIF fCM in Eq. (10) where
the complex dielectric constants (ε0 , ε00 ) are evaluated us-
ing Eqs. (8) and (9) based on the widely used Clausius–
Mossotti model for cubic crystal. In Fig. 3, the inset shows
a structure of DIF below 100 eV, and the shape and the
general behavior of the theoretical results (“—”) are very
similar to the experimental cross sections for solid gold
system, particularly in the low energy structure region.
The present example shows that the alternative expression in Eq. (6) is a useful mean to generate photoionization cross sections of high density system by transforming
those of ideal gas using a DIF function, and that the differences between solid phase and ideal gas cross sections are
from the interactions among particles. The differences on
No. 1
Theoretical Studies on Photoionization Cross Sections of Solid Gold
magnitude between present theoretical results (“—”) and
the experimental photoionization cross sections (“ooo”)
for solid gold at low energy region in Fig. 3 might be partly
caused by the errors embedded in the classical CM model
and the fact that some important quantum effects of solid
phase are not included in the model.
163
measured data (“ooo”) in both the shape and the magnitude even though the calculated results are higher than
the experimental ones at lower energies. The deviations
observed may partly originate from experimental errors
introduced in the deconvolution procedure and from theoretical approximations used in the CM model.
To improve the accuracy of theoretical photoionization
cross sections of a solid system, other physical effects such
as the electron overlap effects may be implemented to the
CM model. For example, the dipole induced-dipole interactions (DID) among the atoms of a cubic crystal can
be inserted into the CM model using the DRF (Direct
Reaction Field) theory,[20]
X
4πχ(1) mac
Eplocal = E mac +
Tpq µq +
E
,
(11)
3
p6=q
Fig. 3 Total photoionization cross sections of gold atom:
“ooo”, the solid gold experimental data; “· · · · · ·”, the ideal
gas experimental results; “—”, the present theoretical studies of solid gold. The curve in the inset is the dielectric
influence function of solid gold.
4 Discussions
In this study, new expressions in Eqs. (3) and (6)
proposed recently[16] have been used to study the photoionization cross sections of solid gold. The new equations are only based on the experimental Beer–Lambert’s
law without using any mathematical approximations and
physical models, and are the correct functions of macroscopic properties of particle number density N and the
complex dielectric constants (ε0 , ε00 ), which implicitly measure the particle interactions of the high density photoionized system and the polarization effects of surrounding
(environmental) polarized particles on the photoionized
atom/molecule. Equation (6) indicates that the photoabsorption (photoionization) cross sections of atoms in a real
system can be obtained by transforming the cross sections
of the same atoms in ideal gas using the DIF. If one has
the complex dielectric constant (ε0 , ε00 ) of a high density
system, the photoionization cross section of the system
can also be directly obtained from Eq. (3).
The frequency and density-dependent photoionization
cross sections of the gold atoms in solid phase show that
density effects are quite important in photoionization experiments of a high density system. The theoretical cross
sections of solid gold using present method agree with the
where Tpq is a modified dipole field tensor, µq is the isolated atomic dipole polarization, which contributes to the
atomic polarizability αp (η, ζ). Then the effective polarizability αp of atom p which is influenced by the µq of atom
q in Eq. (11) can be obtained by solving a set of linear
equations,[20]
X
4πχ(1) αp E mac = cp 1 +
Tpq αp +
,
(12)
3
p6=q
where the constant factor cp will be explained below.
Therefore, one may substitute the effective new polarizability αp (η, ζ) into Eqs. (8) and (9) to obtain the dielectric constants ε0 , ε00 , and then evaluate photoionization
cross sections using Eq. (3) or Eq. (6). The dipole field
tensor Tpq in Eq. (11) can be expressed as
Tpq =
T
E
3fpq
(r̂pq : r̂pq ) − fpq
,
3
rpq
(13)
where rpq is the distance vector between the interactT
ing dipoles (or atoms). The screening functions fpq
and
E
fpq take the effect of overlapping charge densities into
account.[20] In the exponential density model[20] they are
1
E
2
τpq
+ τpq + 1 exp(−τpq ) ,
(14)
fpq
=1−
2
3
τpq
T
E
fpq
= fpq
−
exp(−τpq ) ,
(15)
6
srpq
τpq =
.
(16)
(cp cq )1/6
The empirical screening factor s, and the constant factor
cp , cq are usually optimized to give as good a description
of the average interactions between atoms p and q. Taking
above considerations, the influence of electron cloud overlap effects among atoms on photoionization cross sections
of a high density system can be included in the theoretical
studies.
164
MA Xiao-Guang, SUN Wei-Guo, and CHENG Yan-Song
References
[1] S. Datz, G.W.F. Drake, and T.F. Gallagher, Rev. Mod.
Phys. 71 (2001) S223.
[2] B. Wannberg, H. Veenhuizen, and K.E. Norell, J. Phys.
B19 (1986) 2267.
[3] Rubio Angel, J.A. Alonso, and X. Blasé, Phys. Rev. Lett.
77 (1996) 247.
[4] I. Kanik, L. Beegle, and C. Noren, Chem. Phys. Lett. 279
(1997) 297.
[5] T.C. Jansen, M. Swart, and L. Jensen, J. Chem. Phys.
116 (2002) 3277.
[6] O. Yenen, K.W. Mclaughlin, and D.H. Jaecks, Phys. Rev.
Lett. 86 (2001) 979.
[7] A. Lagendijk, B. Nienhuis, and B.A.V. Tiggelen, Phys.
Rev. Lett. 79 (1997) 657.
[8] J.C. Liu, H.K. Liu, and J.J. Guo, Science in China (series
A) 42 (1999) 1081.
[9] P. Debye, Polar Molecules, Chemical Catalog, Dover Pub.
Inc., New York (1929).
Vol. 43
[10] W.F. Chan, G. Cooper, and C.E. Brion, Phys. Rev. A44
(1991) 186.
[11] U. Fano and J.W. Cooper, Rev. Mod. Phys. 40 (1968)
441.
[12] W. Sun, P.M. Ritzer, and C.W. McCurdy, Phys. Rev.
A40 (1989) 3669.
[13] W. Sun, P.M. Ritzer, and C.W. McCurdy, Science in
China (series A) 40 (1997) 432.
[14] K.W. Meyer and C.H. Greene, Phys. Rev. A50 (1994)
R3573.
[15] C.N. Liu and A.F. Starace, Phys. Rev. A59 (1999) R1731.
[16] W. Sun, X. Ma, and Y. Cheng, Phys. Lett. A362 (2004)
243.
[17] B.L. Henke, E.M. Gullikson, and J.C. Davis, At. Data
and Nuc. Data Tables 54 (1993) 181.
[18] L.D. Landau and E.M. Lifshitz, Statistical Physics,
Addison-Wesley (1958).
[19] R. Haensel, K. Radler, B. Sonntag, and C. Kunz, Solid
State Commun. 7 (1969) 1495.
[20] B.T. Thole, Chem. Phys. 59 (1981) 341.